Su	ei	Petros	kai	epi	taute	petra	oikodomeo	mou	ekklesia
You	are	Peter	and	upon	this	rock	I will build	my	church

Jesus' Petros-petra pun is poetic-prophetic in tone, and must have sounded great, but is this really Jesus talking? When did he ever show a trace of wit, or make a clever pun? He might have spoken these words, but the evidence below will suggest that "Matthew" probably wasn't repeating Jesus' actual words, but instead was writing what he wished Jesus had said for the Greek audience of Matthew's time. As long as his proselytizing was successful there, at that time, he may not have cared whether more literate, critical readers elsewhere, centuries later would figure out what he was up to. But, our case against Matthew and the punning verse does not rest on the unbelievability of a pun in the mouth of the son of God; there is much more, as we shall show below.

As evidence that Jesus may not have spoken the pun elevating Peter to the head of his church, we may note that Matthew never again mentions the extremely important fact that Peter is the head of Jesus' church and heaven's gatekeeper.  It is curious that Mark, who allegedly was Peter's companion and interpreter, reports Simon's confession, but not the crucial "rock and keys" speech by Jesus (Mark 8:27-33); if it had happened, he surely would have mentioned it. 

It is similarly odd that Luke and the authors of the Gospel of John say nothing about it, either. Even Simon Peter himself never mentions a word of it to anyone. As Wheless points out, Paul says that he often defied Simon Peter: "I withstood him to his face," (Galatians 2:11) but in all their disputes, over matters of the faith and the fortunes of the new "church", not a single one of the Apostles stepped forward to put Paul in his proper place by pointing out that Simon Peter was Jesus' hand-picked successor.  Which is more likely:  Matthew's rock and keys passage is fictional, or this really happened, but Matthew was the only one who knew about it?

It is doubtful that Jesus made Peter to the head of his church and awarded him the keys to the kingdom of heaven, because just before Jesus tells Simon he is giving him the keys to heaven, Matthew has Jesus implying that the reason he was awarding Simon Peter the keys was that Jesus' divinity was revealed to him by God:

"And Simon Peter answered and said, Thou art the Christ, the Son of the living God". And Jesus answers, "Blessed thou art, Simon: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven. And I say unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven."  (Matthew 16:15-19)

But, what was so special about Simon's statement? That Jesus was the son of God was common knowledge. The Bible has many earlier references to Jesus as the Christ, or son of God. Indeed, Mark, who said nothing about the rock and keys, says that the following happened before Simon's confession: "And there was in their synagogue a man with an unclean spirit; and he cried out, Saying, Let us alone; what have we to do with thee, thou Jesus of Nazareth? art thou come to destroy us? I know thee who thou art, the Holy One of God." (Mark 1:21-23)

Given this earlier recognition of Jesus' messiahship in Mark, it's no wonder that Mark in his later chapter did not make Jesus see signs of revelation in Simon's confession. Matthew, however, implies that Jesus felt that Peter's recognition had come in a special revelation from God and that he thus merited the great reward which Catholic Encyclopedia (xiii, 261) says Jesus bestowed upon him "for the original(!) and inspired discovery." But, many others had already conceded that Jesus was the Christ, so the far-fetched implications in Matthew 16:17-19 stand as further circumstantial evidence that Matthew's rock and keys story is fictional.